Explanation Given information: If the general quadratic equation x 2 + b x + c = 0 has roots r 1 and r 2 , then c = r 1 r 2 and b = − ( r 1 + r 2 ) Formula used: Quadratic Formula: For a Quadratic equation a x 2 + b x + c = 0 , the value of x is given by: x = − b ± b 2 − 4 a c 2 a Proof: First consider the general equation as: x 2 + b x + c = 0 ..... ( 1 ) Now use the Quadratic Formula to solve the value of x as: x = − b + b 2 − 4 c 2 and x = − b − b 2 − 4 c 2 In the above quadratic equation, a =1. Now, sum of the roots is: r 1 + r 2 = − b + b 2 − 4 c 2 + − b − b 2 − 4 c 2 = − b 2 + b 2 − 4 c 2 − b 2 − b 2 − 4 c 2 = − b − b 2 r 1 + r 2 = − b Therefore, the sum of the roots is the negative coefficient of x The product of the roots is: r 1 r 2 = ( − b + b 2 − 4 c 2 ) ( − b − b 2 − 4 c 2 ) = ( ( − b ) + b 2 − 4 c ) ( ( − b ) − b 2 − 4 c ) 4 Use the formula ( a + b ) ( a − b ) = a 2 − b 2 to simplify the above equation as r 1 r 2 = ( − b ) 2 − ( b 2 − 4 c ) 2 4 = b 2 − ( b 2 − 4 c ) 4 = b 2 − b 2 + 4 c 4 r 1 r 2 = c Therefore, the product of the two roots is constant term Now, consider equation x 2 − 2 x − 8 = 0 .....

7th Edition

ISBN: 9781305115545

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage Learning

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Chapter 1.4, Problem 94E

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To prove:

The relationship between roots and coefficients of a general equation and satisfy the same relationship for the given equations x2−2x−8=0 and x2+4x+2=0.

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Chapter 1.4, Problem 93E

Chapter 1.4, Problem 95E

Chapter 1 Solutions

College Algebra

Ch. 1.1 - The point that is 2 units to the left of the...Ch. 1.1 - If x ¡s positive and y is negative, then the point...Ch. 1.1 - The distance between the points (a, b) and (c, d)...Ch. 1.1 - The point mid ay between (a, b) and (c, d)...Ch. 1.1 - Points in a Coordinate Plane Refer to the...Ch. 1.1 - Points in a Coordinate Plane Refer to the...Ch. 1.1 - Points ¡n a Coordinate Plane Plot the given points...Ch. 1.1 - Points ¡n a Coordinate Plane Plot the given points...Ch. 1.1 - Prob. 9E Ch. 1.1 - Prob. 10E

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To prove: The relationship between roots and coefficients of a general equation and satisfy the same relationship for the given equations x 2 − 2 x − 8 = 0 and x 2 + 4 x + 2 = 0 .

Solution Summary: The author proves the relationship between roots and coefficients of a general equation and satisfy the same relationship for the given equations.

Concept explainers

Videos

To prove:

The relationship between roots and coefficients of a general equation and satisfy the same relationship for the given equations x2−2x−8=0 and x2+4x+2=0.

Want to see the full answer?

Chapter 1 Solutions

To prove: The relationship between roots and coefficients of a general equation and satisfy the same relationship for the given equations x 2 − 2 x − 8 = 0 and x 2 + 4 x + 2 = 0 .

Solution Summary: The author proves the relationship between roots and coefficients of a general equation and satisfy the same relationship for the given equations.

Concept explainers

Videos

To prove: The relationship between roots and coefficients of a general equation and satisfy the same relationship for the given equations x2−2x−8=0 and x2+4x+2=0.

Want to see the full answer?

Chapter 1 Solutions

To prove:

The relationship between roots and coefficients of a general equation and satisfy the same relationship for the given equations x2−2x−8=0 and x2+4x+2=0.